Rota-Baxter algebra and related structures

该项目继续研究Rota-Baxter代数的应用和理论，以及数学和物理之间相互作用产生的相关结构。本项目应用部分追求对微扰量子场论重整化的数学理解，以及重整化方法在数学对象（如发散多重zeta值和Riemann积分）研究中的应用。并结合相关环型的Todd类，研究凸锥的多变量特殊值及其重整化。将在Yang-Baxter方程和可积系统的背景下研究Rota-Baxter算子的一种相对推广称为o算子。该项目的理论部分考虑了这些结构的分类、表示和链条件，以及Rota-Baxter相关算子、shuffle型产品和特殊算子（如树形、pre-Lie和PostLie算子）之间的关系。Rota-Baxter算子是分析中积分算子的代数推广。这个算子的研究是由数学家独立进行的，他们的动机来自概率论，而物理学家则是作为经典杨-巴克斯特方程的解。随后在物理学和数学的几个领域发现了进一步的重要应用，包括量子场论、杨-巴克斯特方程、数论、操作数和霍普夫代数。Rota-Baxter算子的高度递归的树形定义也引起了人们对它与计算数学和组合学的联系的兴趣。对该作业者的进一步研究将有助于了解它在这些不同领域中所起的作用。

This project continues the study of the application and theory of Rota-Baxter algebra and related structures that have arisen from the interaction between mathematics and physics. The application part of this project pursues the mathematical understanding of the renormalization of perturbative quanturm field theory and the application of the renormalization method in the study of mathematical objects, such as divergent multiple zeta values and Riemann integrals. Multivariable special values of convex cones and their renormalization will also be studied in connection with the Todd class of the related toric varieties. A relative generalization of the Rota-Baxter operator called the O-operator will be studied in the context of Yang-Baxter equations and integrable systems. The theoretical part of the project considers the classification, representation and chain conditions of these structures, and the relationship among Rota-Baxter related operators, shuffle type products and special operads such as the dendriform, pre-Lie and PostLie operads. The Rota-Baxter operator is an algebraic generalization of the integration operator in analysis. The study of this operator was independently carried out by mathematicians, with motivation from probability theory, and by physicists, as solutions of the classical Yang-Baxter equation. Further important applications were subsequently found in several areas of physics and mathematics, including quantum field theory, Yang-Baxter equations, number theory, operads and Hopf algebras. The highly recursive, tree-like definition of the Rota-Baxter operator has also generated interest in its connection with computational mathematics and combinatorics. Further study of this operator will contribute to the understanding of the role it plays in such diverse areas.